C₇×#7260 #13184476

order	1792
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₄²×C₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₇×#7260

Subgroups

	subgroup	G/H
center	C₂×C₄×C₇	#102
commutator	C₂×C₈	C₄²×C₇
Frattini, cosocle	C₂³×C₈	C₂²×C₇
socle	C₂²×C₇	#340
prime radical	C₂⁴×C₇	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂³×C₄×C₇ ◁ C₂³×C₇×C₈ ◁ G

336 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 47
transitive degree	μ_t(G) = 448

A minimal faithful permutation representation — 2 generators on 47 points:

(1,2,3,4,5,6,7)(8,9)(10,12)(11,14)(13,15)(16,17,20,25,22,26,21,36)(18,27,29,43,30,44,37,41)(19,31,33,42,35,46,34,24)(23,32,38,45,40,47,39,28);(9,12,14,13)(16,18)(17,23)(19,32)(20,29)(21,37)(22,30)(24,41)(25,38)(26,40)(27,31)(28,34)(33,45)(35,47)(36,39)(42,43)(44,46)

Conjugacy classes & character table

406 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1792, 248);